Financial volatility, Levy processes and power varition

Examples of infinite activity L´evy processes Gamma process The Poisson process and the compound process are by far the most known non-negative L´evy processes.. Inverse Gaussian process

Financial volatility, L´evy processes and power variation

Ole E Barndorff-NielsenThe Centre for Mathematical Physics and Stochastics (MaPhySto),University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark

oebn@mi.aau.dkNeil ShephardNuffield College, Oxford OX1 1NF, UKneil.shephard@nuf.ox.ac.uk

Our papers on this subject are all available at

www.levyprocess.org

This book mauscript is incomplete, comments are very welcome

June 2002

2.1 What is this Chapter about? 8

2.2 What is a L´evy process? 8

2.2.1 The random walk 8

2.2.2 Brownian motion 9

2.2.3 Infinite divisibility 9

2.2.4 The definition of a L´evy process 10

2.3 Processes with non-negative increments — subordinators 10

2.3.1 Examples of L´evy processes 10

2.3.2 L´evy measures for non-negative processes 18

2.3.3 L´evy-Khintchine representation for non-negative processes 20

2.4 Processes with real increments 21

2.4.1 Examples of L´evy processes 21

2.4.2 L´evy-Khintchine representation 30

2.5 Time deformation, chronometers and subordinators 30

2.5.1 Definitions 30

2.5.2 Examples 31

2.6 Quadratic variation 33

2.6.1 Definition and examples 33

2.6.2 Realised variance process 34

2.7 L´evy processes and stochastic analysis 36

2.7.1 Stochastic integrals 36

2.7.2 L´evy-Ito representation of L´evy processes 36

2.7.3 Quadratic Variation 37

2.7.4 Stochastic exponential of a L´evy process 38

2.8 Multivariate L´evy processes 38

2.8.1 Overview 38

2.8.2 Example: multivariate generalised hyperbolic L´evy process 39

2.8.3 Quadratic covariation 40

2.9 Conclusion 41

2.10 Appendix of derivations and proofs 41

2.11 Exercises 43

2.12 Bibliographic notes 43

2.12.1 L´evy processes 43

2.12.2 Flexible distributions 43

2.12.3 L´evy processes in finance 44

2.12.4 Empirical fit of L´evy processes 45

3 Simulation and inference for L´evy processes 47

3.1 What is this Chapter about? 48

3.2 Simulating L´evy processes 48

3.2.1 Simulation 48

3.2.2 Simulating the paths by rejection in the tempered stable case 49

3.2.3 Simulating the paths via the inverse tail integral 50

3.2.4 Simulation via the characteristic function 52

3.3 Empirical estimation and testing of L´evy processes 52

3.3.1 A likelihood approach 52

3.3.2 Model misspecification: robust standard errors 55

3.3.3 Empirical results 57

3.3.4 Olsen scaling rule 60

3.3.5 Fitting multivariate models 62

3.4 Conclusion 66

3.5 Appendix 66

3.5.1 Maximum likelihood estimation of GIG models 66

3.6 Exercises 68

3.7 Bibliographic notes 68

3.7.1 Simulation of L´evy processes 68

3.7.2 Empirical fit of L´evy processes 68

4 Time deformation and chronometers 70 4.1 What is this Chapter about? 71

4.2 General time deformation 71

4.2.1 Introduction 71

4.3 Time deformed Brownian motion 72

4.3.1 Mixture of normals 72

4.3.2 Cumulant functions of y1 73

4.4 Non-negative stationary processes 74

4.4.1 OU type processes 74

4.4.2 Non-negative diffusions 84

4.4.3 Superpositions 87

4.4.4 Higher order autoregressive models 91

4.4.5 General linear models 92

4.5 Integrated non-negative processes 92

4.5.1 General case under covariance stationarity 92

4.5.2 Increments of integrated non-negative processes 95

4.5.3 intOU processes 96

4.5.4 Integrated diffusion based models 100

4.5.5 Superposition of integrated non-negative processes 100

4.6 Conclusion 102

4.7 Appendix 102

4.7.1 Conditions for the existence of an OU process 102

4.8 Exercises 103

4.9 Appropriate literature 103

4.9.1 Time deformation 103

4.9.2 OU type processes 103

4.9.3 Integrated processes 104

5 Stochastic volatility 105

5.1 What is this Chapter about? 106

5.2 Univariate stochastic volatility 106

5.2.1 Basic model 107

5.2.2 SV models and stochastic analysis 110

5.2.3 Leverage 110

5.2.4 Specific results for OU based SV models 111

5.2.5 Specific results for diffusion based SV models 112

5.2.6 SV models with added jumps 112

5.2.7 L´evy processes with SV effects 112

5.2.8 Stationary SV models 112

5.2.9 Econometrics of SV models on low frequency data 112

5.2.10 Empirical performance of SV models on low frequency data 112

5.3 Multivariate stochastic volatility 112

5.3.1 Introduction 112

5.3.2 Factor models 113

5.3.3 Quadratic covariation of SV models 113

5.3.4 Econometrics of multivariate SV models on low frequency data 113

5.4 L´evy based SV models 113

5.4.1 Time deformed L´evy processes 113

5.5 Conclusion 114

5.6 Appendix of derivations and proofs 114

5.7 Exercises 114

5.8 Appropriate literature 114

5.8.1 Stochastic volatility 114

6 Realised variation and covariation 116 6.1 What is this Chapter about? 117

6.2 What is realised variance and covariation? 117

6.2.1 Introduction 117

6.2.2 Probability limits and semimartingales 119

6.2.3 A stochastic volatility model 120

6.3 Asymptotic distribution of realised variance 123

6.3.1 Results and comments 123

6.3.2 Intuition about the result 125

6.3.3 Asymptotically equivalent results 126

6.3.4 Log transforms and realised volatilities 126

6.4 Empirical examples of realised volatilities 128

6.4.1 A time series of daily realised volatilities 128

6.4.2 A time series of annual realised volatilities 128

6.5 Theory and proof of asymptotics for realised variance ∗ 130

6.5.1 A theory and a lemma 130

6.5.2 Proofs 132

6.6 Distribution theory for realised covariation 134

6.6.1 Results and comments 134

6.6.2 Discussion 137

6.6.3 Distribution theory for derived quantities 139

6.7 Empirical example of realised covariation 143

6.8 Theory and proofs of the asymptotics for realised covariance∗ 144

6.8.1 Setting 144

6.8.2 Higher order variations of semimartingales 144

6.8.3 Results 145

6.8.4 Proofs of theorems 147

6.9 Time series of realised variances 150

6.9.1 Framework 150

6.9.2 Model based approach 157

6.10 Conclusion 161

6.11 Bibliographical information 162

6.11.1 Realised variance and empirical finance 162

6.11.2 Quadratic variation, realised variance and econometrics 162

6.11.3 Quadratic covariation 163

6.11.4 Model based estimation of integrated variance 163

7 Power variation 164 7.1 What is this Chapter about? 165

7.2 Introduction 165

7.3 Models, notation and regularity conditions 165

7.4 Results 167

7.5 Proofs 169

7.6 Examples 175

7.7 A Monte Carlo experiment 175

7.7.1 Multiple realised power variations 175

7.7.2 Simulated example 176

7.8 Conclusions 180

7.9 Bibliographical information 181

7.10 Generalising results on realised power variation 182

7.10.1 Stable innovations 182

8 Conclusions 186 A Primer on stochastic analysis 187 A.1 Introduction 188

A.2 Bounded variation 189

A.3 Semimartingales and stochastic integrals 189

A.4 Quadratic variation 191

A.5 Ito’s formula 192

A.6 Stochastic differential equations 193

A.7 Stochastic exponentials 194

A.8 The likelihood ratio process 194

A.9 Girsanov-Meyer Theorem 195

A.10 Multivariate versions 197

A.11 Ito algebra 198

A.12 Results for L´evy processes 198

A.12.1 Types of L´evy processes 198

A.12.2 Stochastic integration 199

A.12.3 L´evy-Ito formula for L´evy processes 199

A.12.4 Quadratic variation of L´evy processes 199

A.12.5 Density transformations 200

B Collections of definitions and notation 201

B.1 Motivation 202

B.2 Notation 202

B.3 Distributions 209

B.3.1 Generalised inverse Gaussian (GIG) distributions 209

B.3.2 Generalised hyperbolic (GH) distributions 212

B.3.3 Stable based distributions 215

Chapter 1

Introduction

Chapter 2

Abstract: This Chapter provides a first introduction to the use of L´evy processes as models oflog-prices in financial markets, focusing on the probabilistic aspects Univariate and multivariatemodels are discussed A detailed bibliographical review is given at the end of the Chapter.

It is important to keep in mind that L´evy processes allow a flexible model for the marginaldistribution of returns, but still maintain that returns are iid Obviously this is a very poordescription of reality and later on in our book we extend this framework to allow for stochasticvolatility However, a significant understanding of these processes does help our understandingboth of the economics and the development of the later techniques given in the book

2.1 What is this Chapter about?

In this Chapter we provide a first course on L´evy processes in the context of financial economics.The focus will be on probabilistic and econometric issues; understanding the models and theirfit to returns on speculative assets We leave until our Second book the vital issue of how thesemodels can guide investors in their the allocation of resources between risky and riskless assetsand the pricing of derivatives written on L´evy processes The Chapter will refer to some commondatasets discussed in detail in Chapter 1 and will delay the discussion of literature on this topicuntil the end of this Chapter Throughout we hope our treatment will be as self-contained aspossible At the end of this book we have given a brief introduction, called “The Primer,” tostochastic analysis which may be of help to readers without a strong background in probabilitytheory

This long Chapter has 8 other sections, whose goals are to:

• Introduce L´evy processes with non-negative increments

• Extend the analysis to L´evy processes with real increments

• Introduce time deformation, or time change, where we replace calendar time by a randomclock

• Introduce quadratic variation, a central concept in econometrics and stochastic analysis

• Brief discuss stochastic analysis in the context of L´evy processes

• Introduce various methods for building multivariate L´evy processes

• Draw conclusions to the Chapter

• Discuss the literature associated with L´evy processes

This Chapter leads into the next one, which will focus on methods for simulating the paths

of L´evy processes and the estimation and testing of these models on financial time series

2.2 What is a L´ evy process?

The most basic model of the logarithm of the price of a risky asset is a random walk It is built

by summing independent and identically distributed (i.i.d.) random variables c0, c1, to deliver

The process is written in discrete time and is moved by the i.i.d increments

Hence future changes in a random walk are unpredictable

Random walks live in discrete time What is the natural continuous time version of thisprocess? There are at least two strong answers to this question

to a scaled version of Brownian motion, as T goes to infinity At first sight this suggests the onlyreasonable continuous time version of a random walk, which will sum up many small events, isBrownian motion This insight is, however, incorrect

2.2.3 Infinite divisibility

Our book follows a second approach Suppose that the goal is to design a continuous timeprocess at time 1, z(1), which has a distribution D It may be possible to divide the timefrom zero until one into T pieces, each of which has independent increments from a commondistribution D(T ) such that the sum

c(T )s , where c(T )s i.i.d.∼ P o(1/T ),

this produces a valid random walk due to the fact that the independent Poisson incrementssum to a Poisson Hence this process makes sense even as T goes to infinity and so this type

of construction can be used as a continuous time model — the Poisson process The class ofdistributions for which this construction is possible is those for which D is infinitely divisible.The resulting processes are called L´evy processes Examples of infinitely divisible distributionsinclude, focusing for the moment on only non-negative random variables, the Poisson, gamma,reciprocal gamma, inverse Gaussian, reciprocal inverse Gaussian, F and positive stable distri-butions

2.2.4 The definition of a L´evy process

The natural continuous time version of the discrete time increment given in (2.1) is, for anyvalue of ∆ > 0,

z(t + ∆) − z(t), t ∈ [0, ∞] Increments play a crucial role in the formal definition of a L´evy process

Definition 1 L´evy process The stochastic process

z(t), t ∈ [0, ∞] , z(0) = 0,

is a L´evy process if and only if it has independent and (strictly) stationary increments

In the definition the first assumption means that the shocks to the process are independentover time and that they are summed, while the second assumption means that the distribution

of z(t + ∆) − z(t) may change with ∆ but does not depend upon t The independence andstationarity of the increments of the L´evy process means that

C {θ ‡ z(t)} = log [E exp {iθz(t)}]

= t log [E exp {iθz(1)}]

in-2.3 Processes with non-negative increments — subordinators

2.3.1 Examples of L´evy processes

Motivation

We start with L´evy processes with non-negative increments Such processes are often calledsubordinators This is our focus for two reasons: (i) they are mathematically considerablysimpler, (ii) most of models we build in this book will have components which are L´evy processeswith non-negative increments and so they are a major concern to us The discussion of processes

on the real line will be given in the next section In order to reduce the technical demands onthe reader we are mostly going to use cumulant functions in this context as this is sufficient forour purposes As the processes are positive, it is natural to work with the kumulant function inthe form

K {θ ‡ z(1)} = log [E exp {−θz(1)}] , where θ ≥ 0

Occasionally the more standard cumulant function

K {θ ‡ z(1)} = log [E exp {θz(1)}] , where θ ≥ 0,will be used

Poisson process

Introduction Suppose we count the number of events which have occurred from time 0 until

t ≥ 0 The very simplest continuous time model for this type is a L´evy process with independentPoisson increments

z(1) ∼ P o(ψ), ψ > 0,with density

(b) Simulated CPP with I G (1,1) innovations

Figure 2.1: (a) Sample path of a homogeneous Poisson process with intensity ψ = 1 Horizontalaxis is time t, vertical is z(t) (b) Corresponding compound Poisson process with cs∼ IG(1, 1).code: levy graphs.ox

For the Poisson process

one, right continuous

lim

s↓tz(s) = z(t)and has limits from the left

z(t−) = lim

s↑tz(s)

For such processes the jump just before time t is written as

∆z(t) = z(t) − z(t−)

This notation clashes with our use of ∆ to stand for a time interval

We might expect these types of jumps to appear in financial processes due to dividendpayments or news, such as macroeconomic announcements A process with this mathematicalproperty is called a c`adl`ag (continue `a droit, limite `a gauche) or a RCLL (right continuousleft limit) process in the literature It causes no restrictions as regard the finite dimensionaldistributions and we follow the convention here that all L´evy processes are c`adl`ag, unless other-wise stated The similiarly named property c`agl`ad (continue `a gauche, limite `a droit) plays animportant role in our Appendix on stochastic analysis

Poisson process is a special semimartingale Semimartingales play a central role in ern stochastic analysis, in particular providing a basis for the definition of a stochastic integral.Consequently it is important to note that a Poisson process is a semimartingale We see this bynoting that any semimartingale x can be decomposed into

In the case of the Poisson process we can see that z can be decomposed in this way by writing

a(t) = E {z(t)} = ψt and m(t) = z(t) − ψt

The m process, which is often called a compensated Poisson process, is a mean-0 martingale,for E(m(t)|Ft) = 0 It will turn out that all L´evy processes for whom E {z(t)} exists are specialsemimartingales This is an important result

As z is a L´evy process the theory of semimartingales implies that if h is locally squareintegrable then the stochastic integral

y(t) =

Z t

0 h(u−)dz(u),can be constructed This is often written in the more abstract notation as the process

where 0 = u0 < u1 < < un= t Any such integral process y is itself a semimartingale Moreformal details of stochastic integrals are given in our Primer When z is a Poisson process ysimplifies to the random sum

where τ1, τ2, , τz(t) are the arrival times of z

In the case of a special semimartingale we can view da(t) informally as the continuouslyupdated expected return conditional on the information just before t, while dx(t) is the returnand dm(t) is the unanticipated return Without predictability on a(t) this decomposition isnot possible Thus, from an economic viewpoint special semimartingales seem of fundamentalimportance

Ito formula for a Poisson process Suppose we wish to look at a function of some stockprice process s, y(t) = f (s(t)) where s(t) = µt + z(t), which is drift plus a Poisson process

z This is a continuous time version of a binomial tree model frequently used in financialeconomics1 Ito’s formula for semimartingales given in (A.11) applies here, for all L´evy processesare semimartingales In this case

{f {s(u)} − f {s(u−)} − f0{s(u−)} ∆u}, (2.4)

which can be written in the form of a stochastic differential equation (SDE) as

dy(t) = f0{s(t−)} ds(t) +£f {s(t)} − f {s(t−)} − f0{s(t−)} ∆s¤.This can be simplified as ds(t) = µdt + ∆z(t) and ∆s(t) = ∆z(t) to

dy(t) = f0{s(t−)} µdt + f {s(t)} − f {s(t−)} The above analysis is interesting for it means that the portfolio V = y − δs, where δ =

f {s(t−) + 1} − f {s(t−)} has the important property that, if a riskless interest rate r exists,then

dV = dy − δds = f0{s(t−)} µdt

is instantly riskless and so the porfolio must grow at the riskless rate dV = rV dt As thisargument does not need us to specify a utility function we can price y as if all agents were riskneutral (Cox and Ross (1976)) and so price contingent assets not under s but under the riskneutral process rt + {z(t) − Ez(t)} This argument is simply a continuous time version of astandard binomial tree, for it relies on knowing that the value of the portfolio will either jump

or not jump by one unit It implies that familiar Black-Scholes analysis, based on Brownianmotion and hedging, is not upset by jumps of known size

Equivalent martingale measure (EMM) An equivalent measure is defined as a measure Qt

such that the Radon-Nikodym derivative dQt/dPt is strictly positive, where Ptis the measure ofthe original z process from time 0 up to time t It implies that the support of the process under

Qtmust be the same as the original process for the z process This means Q must correspond tothe measure generated by a counting process to be an equivalent measure However, equivalencedoes not constrain the intensity of the counting process and so there are an infinite number ofequivalent measures for this problem This is standard in financial economics

1 This exposition was suggested to us by George Konaris, who we thank for allowing us to use it here.

In order for Qtto be an equivalent martingale measure (EMM) we additionally need

so the intensity will increase with the level of the Poisson process

Compound Poisson process

Introduction Suppose {N(t)}t≥0 is a Poisson process and {cs} is an i.i.d sequence Thendefine a compound Poisson process as

Here cN (t) is the random innovation of the compound Poisson process.

When we construct a portfolio V = y − δs it is not possible to find a δ to make dV istic for cN (t) is not known at time t− This is a crucial observation for it implies any compoundPoisson process yields an incomplete financial market and prices of contingent claims cannot

determin-be determined simply using hedging Some introduction of a utility function will determin-be necessary.This will be discussed in detail in our second book

Examples of infinite activity L´evy processes

Gamma process The Poisson process and the compound process are by far the most known non-negative L´evy processes The jumps happen, typically, rather rarely Consequentlyincrements to these processes are often exactly zero, even when measured over quite large timeintervals This feature of the process is fundamentally different from the gamma L´evy process

well-A gamma L´evy process z makes z(1) obey a gamma law

z(1) ∼ Γ(ν, α), ν, α > 0,with density

¶

,

which implies z(t) ∼ Γ(νt, α) The gamma process has the useful property that it has incrementswhich are strictly positive whatever small time interval has elapsed Such L´evy processes aresaid to have infinite activity This feature puts them apart from a compound Poisson process

A sample path of a gamma process is drawn in Figure 2.2(a) The path is not continuous(anywhere) It was drawn by splitting time into intervals of length 1/2000 and sampling fromthe implied random walk with Γ(ν/2000, α) distribution Very similar paths are produced byusing smaller time intervals The process is a rough upward trend with occasional large shifts

Inverse Gaussian process An inverse Gaussian (IG) process z requires the L´evy process

at time one to follow an inverse Gaussian distribution z(1) ∼ IG(δ, γ), where δ > 0, γ ≥ 0, withdensity

of time Most of these jumps are tiny, while occasionally there are larger upwards movements.The form of the cumulant function implies z(t) ∼ IG(tδ, γ)

(b) Simulated I G (0.2,100) Lévy process

Figure 2.2: Simulated Γ and IG L´evy processes, using intervals of length 1/2000 Codelevy graphs.ox

A sample path of an IG L´evy process is drawn in Figure 2.2(b) The parameters were selected

to have the same mean and variance of z(1) as that used to draw the path of the gamma processgiven in Figure 2.2 Again the process is a rough upward trend with occasional large shifts

Some other non-negative processes A reciprocal (inverse) gamma (RΓ) process z requiresthe L´evy process at time one to be a reciprocal gamma variable z(1) ∼ RΓ(ν, α), α, ν > 0,withdensity

we will show later that we can use computationally intensive methods to simulate the process

A lognormal (LN) process z requires the L´evy process at time one to be a lognormal variablez(1) ∼ LN(µ, σ2), σ2 ≥ 0, with density

fz(1)(x) = 1

x√2πexp

A reciprocal (inverse) Gaussian (RIG) process z requires the L´evy process at time one to be

a reciprocal inverse Gaussian variable z(1) ∼ RIG(δ, γ), δ > 0, γ ≥ 0, with density

Similar remarks hold for the positive hyperbolic (PH) process which requires that z(1) ∼

In the special case of δ → 0, then

This distribution is again infinitely divisible but is again difficult to work with

Generalised inverse Gaussian process The above infinite activity processes are all specialcases of the generalized inverse Gaussian (GIG) process This puts

z(1) ∼ GIG(ν, δ, γ),with GIG density

where again Kν(·) is a modified Bessel function of the third kind This density has been shown

to be infinitely divisible and so supports a whole nesting class of L´evy processes Prominentspecial cases are achieved in the following ways:

IG(δ, γ) = GIG(−12, δ, γ), P H(δ, γ) = GIG(1, δ, γ),RΓ(ν, δ2/2) = GIG(−ν, δ, 0), Γ(ν, γ2/2) = GIG(ν > 0, 0, γ),RIG(δ, γ) = GIG(12, δ, γ), P HA(δ, γ) = GIG(0, δ, γ)

RP H(δ, γ) = GIG(−1, δ, γ)

Here all these distributions are familiar except for the positive hyperbola distribution, which isdenoted P HA In order to obtain these results we have to allow δ or γ to be zero 0 In thesecases the GIG’s density has to be interpreted in the limiting sense, using the well-known resultsthat for x ↓ 0 we have

a modelling viewpoint, however we will see that practical modelling will sometimes be carriedout directly via some of the terms which make up the L´evy-Khintchine representation Hence agood understanding of this section is essential for later developments

Poisson and compound processes

To start off with think of a Poisson process, so that z(1) ∼ P o(ψ) Then, writing δ1(x) as theDirac delta centred at x = 1, we write

to be introduced Instead of working with probability measures we will have to use more generalmeasures W concentrated on R+ An important point is that some of the measures that will beused later will not be integrable (that is R∞

0 W (dx) = ∞) and so probability measures are sufficient for a discussion of L´evy processes In the simple Poisson case measures are introduced

in-by expressing

K{θ ‡ z(1)} = −

Z ∞

0 (1 − e−θx)W (dx) (2.9)where W = ψδ1 is called the L´evy measure Of course this measure is integrable, indeed itintegrates to ψ

Let us now generalise the above setup to the compound Poisson process (2.6), but stillrequiring {cs} to be strictly positive — ruling out the possibility that cs can be exactly zerowith non-zero probability Then, writing the distribution function of c1 as P (x ‡ c1),

K{θ ‡ z(1)} = −ψn1 − exp K (θ ‡ c1)o

to the probability measure In the simple case where c1 has a density we write

W (dx) = w(x)dxand call w(x) (which is ψ times the density of c1) the L´evy density In such cases the kumulantfunction becomes

Of course this L´evy density integrates to ψ — not one

Although Poisson and compound processes have integrable L´evy measures (for W is portional to a probability measure which integrates to one) theoretically more general L´evyprocesses can be constructed without abandoning the form (2.9) The non-integrable measures

pro-W will not correspond to compound Poisson processes To ensure that they yield a valid mulant function we require that R∞

ku-0 (1 − e−θx)W (dx) exists, while continuing to rule out thepossibility that W has an atom at zero It is simple to prove that a necessary and sufficientcondition for R∞

up in the following two examples

Example 2 It turns out that the L´evy density of z(1) ∼ IG(δ, γ) is

w(x) = (2π)−1/2δx−3/2exp(−γ2x/2), x ∈ R+ (2.10)This L´evy density is not integrable as it goes off to infinity too rapidly as x goes to zero This

is important for it implies an IG process is not a compound Poisson process Although theL´evy density is not integrable it does satisfy the finiteness condition on R∞

0 min (1, x) W (dx)for the addition of the x factor regularises the density near zero (it behaves proportionally to aΓ(1/2, γ2/2) variable for x ≤ 1)

Example 3 It can also be shown that the L´evy density of z(1) ∼ Γ(ν, α) is

w(x) = νx−1exp(−αx), x ∈ R+ (2.11)Again this is not an integrable L´evy density although it is slower to go off to infinity than theinverse Gaussian case This mean in practice that it will have a smaller number of very smalljumps than the IG process

These two results are special cases of the result for the GIG(ν, δ, γ) probability density (2.8).The corresponding L´evy density is then

w(x) = Cx−1−α, where C = δ2α α

Γ (1 − α), (2.13)while z(t) ∼ P S(α, tδ)

2.3.3 L´evy-Khintchine representation for non-negative processes

Representation

Having allowed the L´evy measure not to be integrable, a single extra step is required in order

to produce a general setup We allow a drift a > 0 to be added to the cumulant function This

is carried out in the following fundamental theorem

Theorem 2.1 L´evy-Khintchine representation for non-negative L´evy processes Suppose z is aL´evy process with non-negative increments Then the kumulant function can be written as

The importance of this representation is that the kumulant function of all non-negativeL´evy processes can be written in this form In other words, non-negative L´evy processes arecompletely determined by a and the L´evy measure W (which has to satisfy (2.15)) No otherfeature is necessary

In the special case whenR∞

0 W (dx) < ∞ we say that z is of finite activity — indeed all suchprocesses can be written as a compound Poisson process In cases where this does not hold, z

is said to be an infinite activity process

Models via the L´evy density: tempered stable process

An important implication of the L´evy-Khintchine representation is that L´evy processes can bebuilt by specifying a and W directly, implying the probability density of z(1) An importantexample of this is the tempered stable, T S(κ, δ, γ), class which tilts the L´evy density of thepositive stable The result is

K{θ ‡ z(1)} = δγ − δ³γ1/κ+ 2θ´ κ

,while the corresponding first two cumulants are

2κδγ(κ−1)/κ and 4κ (1 − κ) δγ(κ−2)/κ.Finally the cumulant function implies the convenient property that z(t) ∼ T S(κ, tδ, γ)

2.4 Processes with real increments

2.4.1 Examples of L´evy processes

z(1) ∼ N(0, 1),with density

K {θ ‡ z(1)} = log [E exp {θz(1)}] = 12θ2.The implication of this is that marginally z(t) ∼ N(0, t), while increments

Figure 2.3: (a) Sample path of √

0.02 times standard Brownian motion with z(0) = 0 (b)Sample path of a NIG(0.2,0,0,10) L´evy process with z(0) = 0 Thus the increments of bothprocesses have the same variance Code: levy graphs.ox

z(t) = µt + σb(t)

∼ N(µt, tσ2),with increments

z(t + ∆) − z(t) ∼ N(µ∆, σ2∆)

The associated cumulant function for z(1) is µθ +12θ2σ2

A graph of a sample path from standard Brownian motion is displayed in Figure 2.3(a) Itillustrates that the path is continuous In a moment we will see that Brownian motion is theonly L´evy process with this property — all other L´evy processes have jumps

and a non-unit variance It was discussed in some detail in the first Chapter of this book This

is an important model in practice for quite a lot of effort has been expended on working on thederivative pricing theory associated with this simple structure

Location scale mixture processes

Normal inverse Gaussian process If we assume σ2 ∼ IG(δ, γ) and ε is an independentstandard normal variable then

z(1) ∼ NIG(α, β, µ, δ), µ ∈ R, δ ∈ R+, 0 ≤ β < αwhich has the density

fz(1)(x) = a(α, β, µ, δ)q

µ

x − µδ

of values of χ, ξ Such a plot is called a shape triangle As ξ → 0 so the log-density becomesmore quadratic, while for values around 0.5 the tails are approximately linear For larger values

of ξ the tails start decaying at a rate which looks appreciably slower than linear In the limit as

ξ → 1 the density becomes a Cauchy variable

This model has recently received considerably attention as a tractable alternative to nian motion as a model of log asset prices One of its advantages is that the resulting cumulantfunction is

Figure 2.4: Shape triangle for the NIG model That is we graph the shape of the log-density forthe NIG model for a variety of values of the steepness parameter ξ and the shape parameter χ.This graph was kindly made available to us by Preben Blæsild.

0.35 (b) Simulated Laplace Lévy process

Figure 2.5: (a) Sample path of a N Γ(4,200,0,0) L´evy process Such processes are often calledvariance gamma processes in the literature (b) Sample path of a La(0.2,0,0) L´evy process.Code: levy graphs.ox

Normal gamma process If we assume σ2 ∼ Γ(ν, γ2/2) and ε is an independent standardnormal variable then

y = µ + βσ2+ σε ∼ NΓ(ν, γ, β, µ),which we will call the normal gamma distribution From the cumulant function

K {θ ‡ z(1)} = µθ + ν log

Ã

1 +θβ + θ

2/2γ

´1−2ν

√2πΓ(ν)2ν−1K¯

Hyperbolic and Laplace processes If we assume σ2 ∼ P H(δ, γ) and ε is an independentstandard normal variable then

y = µ + βσ2+ σε ∼ H(α, β, µ, δ), where α =

q

β2+ γ2,has the hyperbolic distribution This distribution can be shown to be infinitely divisible, al-though the proof of this is difficult The hyperbolic process puts z(1) ∼ H(α, β, µ, δ), where thedensity is

Hyperbolic L´evy processes have the disadvantage that we do not have an exact expressionfor the density of z(t) for t 6= 1, nor can we simulate from the process in a non-intensive manner

Figure 2.6: Shape triangle for the hyperbolic model That is we graph the shape of the density for the hyperbolic model for a variety of values of the steepness parameter ξ and theshape parameter χ This graph was kindly made available to us by Preben Blæsild.

log-Both of these properties are inherited from the fact that this is also the case for the positivehyperbolic process we discussed in the previous section

The Laplace distributions (symmetric and asymmetric) occur as limiting cases of (2.20) for

α, β and µ fixed and δ ↓ 0 We write this as La(α, β, µ) The corresponding density is

Finally, if σ2 ∼ RP H(δ, γ), instead of being positive hyperbolic, then y ∼ RH(α, β, µ, δ),the reciprocal hyperbolic This distribution is again infinitely divisible The RH process puts

fz(1)(x) =

q

α2− β22αδK−1

a skewed Student’s t distribution, which is infinitely divisible and so can be used as the basis of

a L´evy process The skewed Student’s t process puts z(1) ∼ T (ν, δ, β, µ), where the density is

The more familiar Student’s t distribution is found when we let β → 0, then the density becomes

¶ 2 )−ν−1/2

In both cases this process has the interesting feature that only moments of order less than ν willexist — at any time horizon However, we do not know the distribution of z(t) for this process,while simulation has to be carried out in quite an involved manner Hence this process is not aseasy to handle as the N IG or normal gamma L´evy processes

A similar type of complexity occurs if we replace σ2 ∼ RΓ(ν, δ2/2) by a reciprocal inverseGaussian distribution, RIG(δ, γ) The resulting mixture distribution is called a normal recipro-cal inverse Gaussian distribution N RIG(α, β, µ, δ) which can be shown to be infinitely divisibleand so supports a N RIG L´evy process This process is again hard to work with and so we willnot discuss it in detail here

Generalized hyperbolic process If we assume σ2 ∼ GIG(ν, δ, γ) and ε is an independentstandard normal variable then

y = µ + βσ2+ σε ∼ GH(ν, α, β, µ, δ), where α =

q

β2+ γ2,the generalised hyperbolic distribution This distribution includes as special cases the normal(N ), normal inverse Gaussian (N IG), normal reciprocal inverse Gaussian (N RIG), hyperbola(HA), hyperbolic (H), skewed Laplace (La), normal gamma (N Γ) and skewed Student (T )distributions in the following way:

for ν > 0 The generalised hyperbolic distribution is infinitely divisible and so can be used

as the basis of a rather general L´evy process whose special cases obviously include Brownianmotion with drift and the N IG, hyperbolic, hyperbola, N RIG, normal gamma, skewed Laplaceand skewed Student L´evy processes The proof of infinite divisibility of this distribution isinvolved but will be discussed in Part II of our book The generalised hyperbolic process putsz(1) ∼ GH(ν, α, β, µ, δ), where the density is

*

Kν+1(δγ)δγKν(δγ)+

It is helpful to reexpress this density in a new alternative format We do this by defining

Kν(x) = xνKν(x), noting that Kν(x) = K−ν(x) and setting up some scale invariant parameters

µx − µδ

¶¾

Not surprisingly, in general we do not know the GH density of z(t) for t 6= 1, nor can wesimulate from the process in a non-intensive manner This model is so general that it is typicallydifficult to manipulate mathematically and so is not often used empirically Instead special casesare usually employed

Symmetric stable processes One of the most studied L´evy processes is the symmetric stableprocess This puts

z(1) ∼ S(α, δ), 0 < α ≤ 2, δ > 0,

a symmetric stable distribution with index α Except for the boundary case of α = 2, thisdistribution has the empirically unappealing feature that the variance of z(1) is infinity Thedensity of this variable is unknown in general, with exceptions being the Gaussian variable(α = 2), the Cauchy variable (α = 1) and the L´evy variable (α = 1/2) Despite the complexity

of the density the cumulant function is simply

K {θ ‡ z(1)} = δθα,which implies z(t) ∼ S(α, tδ) The L´evy density for a symmetric stable process is given by

Normal tempered stable process If we assume σ2 ∼ T S(κ, δ, γ) and ε is an independentstandard normal variable then

y = µ + βσ2+ σε ∼ NT S(κ, δ, γ, β, µ),

a normal tempered stable distribution, which is infinitely divisible and so can be used as thebasis of a L´evy process This process has as special cases the N IG L´evy process and the normalgamma L´evy process (when κ ↓ 0) This process will be discussed in more detail in a latersection

Some other L´evy processes living on the real line

Truncated L´evy flights If the L´evy density of the stable process is truncated, so that

w(x) =

(

c |x|−1−α for x ∈ [−l, l]

0 otherwise,where c, l > 0, then it can be shown that this again supports a L´evy process where z(1) has afinite variance This process is called a truncated L´evy flights process This has received someinterest in the context of the econophysics literature

Extended Koponen class Another example of a L´evy process specified through the L´evydensity is the KoBoL (after Koponen, Boyarchenko and Levendorskii) or extended Koponenclass This puts

aζ−iβ 2

´





.

The resulting process is obviously infinitely divisible and so supports a L´evy process with z(t) ∼

M eixner(a, b, dt, µt) Further, the L´evy density is

w(x) = d e

bx/a

x sinh(πx/a),and so we can see that this L´evy process has infinite activity Unfortunately we do not know asimple way of simulating from this distribution

2.4.2 L´evy-Khintchine representation

The L´evy-Khintchine representation for positive variables given in (2.14) can be generalised tocover L´evy processes with increments on the real line Three basic developments are needed.First, the L´evy measure must be allowed to have support on the real line, not just the positivehalf-line, but still excluding the possibility that the measure has an atom at zero Second, theparameter a needs to be allowed to be a real variable, not just positive Third, we imaginethat an independent Brownian motion component is added to the process The result is thecelebrated L´evy-Khintchine representation for L´evy processes

Theorem 2.2 L´evy-Khintchine representation Suppose z is a L´evy process Then the log ofthe characteristic function can be written as

L´evy processes are completely determined by the characteristic triplet: a, the variance σ2 ofthe Brownian motion and the L´evy measure W (which has to satisfy (2.26)) No other feature isnecessary and every such triplet¡

in this context is new

Definition 2 A chronometer is any non-decreasing random process The special case where thechronometer has independent and stationary increments is called a subordinator

The requirement that the chronometer is non-decreasing rules out the chance that time can

go backwards A special case of a chronometer is a subordinator, while subordinators are specialcases of L´evy processes (e.g Poisson or IG L´evy processes are subordinators) All subordinatorsare pure upward jumping processes We should note here that the finance literature typicallylabels chronometers subordinators, while the probability literature only discusses deformation

in the context of L´evy processes

In this section we will study what happens when a subordinator is used to change the clock,that is deform, a stochastically independent L´evy process Write v(t) and τ (t) as independentL´evy processes, the latter being a subordinator used to model the random clock The result is

z(t) = v(τ (t))

The increments of this process are

z(t + ∆) − z(t) = v(τ(t + ∆)) − v(τ(t))

= v(τ (t) + {τ(t + ∆) − τ(t)}) − v(τ(t)),which are independent and stationary and so z is a L´evy process

Brownian motion is the only L´evy process with continuous sample paths, however this erty does not survive being deformed by a subordinator

prop-The subordinator must be a pure jump process — jumping upwards at random times Ateach instant of a jump z(t) must (with probability one) also jump, while in instants where thesubordinator does not jump the level of z(t) is left unchanged

Brownian motion with a Poisson subordinator

Assume v(t) ∼ N(βt, σ2t) is a scaled Brownian motion with drift and that it is deformed by aPoisson process with intensity ψ Then

(c) Poisson process subordinator

Figure 2.7: Figure (a) deformed Brownian motion using a Poisson process subordinator Figure(b) path of the Brownian motion Figure (c) Poisson process subordinator Code: levy code.ox

Brownian motion subordinated by IG — the NIG L´evy process

Suppose τ (t) is an inverse Gaussian L´evy process with τ (1) ∼ IG(δ, γ) and v(t) is Brownianmotion with drift β Then z(t) is a L´evy process In particular

z(1)|τ(1) ∼ N(βτ(1), τ(1))and so unconditionally the increments are independent with

z(1) ∼ NIG(α, β, 0, δ), α2 = β2+ γ2.Hence this deformed Brownian motion is a special case of the normal inverse Gaussian L´evyprocess, which we simulated in Figure 2.2(b)

Normal tempered stable L´evy process

Suppose τ (t) is a tempered stable T S(κ, δ, γ) L´evy process, with L´evy density given in (2.16)

It is a subordinator Then if we assume bβ(·) is Brownian motion plus drift and we write

z(t) = µt + bβ(τ (t)),then z is called a normal tempered stable (N T S) L´evy process We write

z(1) ∼ NT S(κ, α, β, µ, δ),but the corresponding probability density is generally unknown (except for an infinite seriesrepresentation, see Feller (1971, p 583)) The cumulant function, on the other hand, is rathersimple

K(θ ‡ z(1)) = µθ + δγ − δnα2− (β + θ)2oκ, where α =

q

β2+ γ1/κ.The form of this function implies

Type G L´evy processes

In the probability literature, L´evy processes which can be written as z(t) = µt + bβ(τ (t)), forsome subordinator τ , which we shall call type G L´evy processes — the subset of L´evy processesfor which there is a deformation of Brownian motion interpretation Many well known L´evyprocesses are not in this class

2.6 Quadratic variation

2.6.1 Definition and examples

A commonly used measure of continuous time processes in financial economics is the QuadraticVariation (QV) process This has two steps First, time is split into small intervals

sup

i {tri+1− tri} → 0 for r → ∞

This series looks at the partial sum of squared increments over tiny intervals of time Ingeneral the QV process of a L´evy process is a (different) L´evy process for the increments areindependent and stationary as QV are just sums the squares of independent and stationaryincrements Further, it can be regarded as a subordinator for the increments are non-negative

To illustrate these points two examples are given

of drift Of course in practice this is highly misleading argument for the continuous time model

is unlikely to be perfectly specified at very short time horizons

Brownian motion deformed by a Poisson process

Suppose z is constructed by time deforming a Brownian motion with drift β with a Poissonprocess τ , then z is a compound process (2.27) and

Hence the QV is also a compound process with [z](t) ∼ χ2

τ (t) Hence [z](t) is a noisy estimator

of τ (t), but they share the same expectation

Semimartingales

We noted in section 3 that all L´evy processes were semimartingales, and that when the mean

of the increments of the process existed then they are special semimartingales This means wecan uniquely write them as

z(t) = a(t) + m(t),

a predictable component of locally bounded variation and a local martingale, respectively It ispossible to show that for all semimartingales

[z](t) = [m](t)

Quadratic variation is very important One of the reasons for this is that for all localsemimartingales with continuous predictable components

Var(dz(t)|Ft) = E (d[z](t)|Ft) ,

so long as the moments exist

2.6.2 Realised variance process

Definition and basics

In applied economics it is often inappropriate to study returns over infinitely small time intervalsfor our models tend to be highly misspecified at that level due to market microstructure effects

In particular the idea of a unique price is a fiction for the transaction price tends to dependupon, for example, the volume of the deal, the reputation of the buyer and seller, prevailingliquidity (and so time of day) and the initiator (i.e was it the buyer or the seller) These issueswill be discussed at more length in later chapters To avoid the worst effects of misspecification,

a finite version of quadratic variation is often used This is called the realised volatility orvariance process This splits time into intervals of length δ and computes the correspondingsum of squares

The realised variance process is defined, for δ > 0,

in time and so has a number of features of a discrete time random walk

A numerical example of the realised variance process is given in Figure 2.8, which computes

it for a N IG L´evy process In this picture we have taken δ = 1 and δ = 1/10, so taking one and

10 squared observations per unit of time, respectively Also given is the corresponding limit, thequadratic variation We see that as δ gets small so the realised variance process becomes a goodapproximation of the QV

In order to understand the connection between the L´evy process, the realised variance processand the quadratic variation it is helpful to think about the following calculation Hold t fixedand set the choice of δ so that M = t/δ, then

Figure 2.8: Figure (a) Sample path of NIG(0.2,0,0,10) L´evy process (b) Sample path of sponding realised variance process taking δ = 1 — one observation per unit of time (c) Samebut with δ = 1/10 — ten observations per unit of time (d) Quadratic variation of the process.Code: levy code.ox.

corre-where κr denotes the r-th cumulant of z(1) The only one of these results which is not forward is

A simple example of this is where z is standard Brownian motion, then κ3 = κ4= 0, whichmeans that

Notice the covariance is singular, for z(t) = [z](t).

A more interesting example occurs when z is scaled Brownian motion which is deformed bythe subordinator τ Then

κ1= 0, κ2= σ2κ1(τ ), κ3 = 0, κ4 = 3σ2κ2(τ ),where κs(τ ) denotes the s-th cumulant of τ (1) Hence in this case

Var h[z](t) − [zδ] (t)i = 3σ4κ21(τ )t2M−1,which only depends upon the mean of the subordinator, not its variance

2.7 L´ evy processes and stochastic analysis

2.7.1 Stochastic integrals

This section will assume a basic knowledge of stochastic analysis — that is the calculus ofstochastic integrals based on semimartingales For those unfamiliar with this background wehave provided a very short primer to this material in Section A

All L´evy processes are semimartingales So, in particular, we can consider stochastic integrals

of the form

f (·, A) • z,where z is a L´evy process, f is a real function on Rx× R and A is a c`agl`ad stochastic process,and f satisfies some mild regularity condition ensuring that the process f (·, A) is again c`agl`ad

2.7.2 L´evy-Ito representation of L´evy processes

Consider first the case of L´evy subordinators It can be shown that any non-negative L´evyprocess z is representable in the L´evy-Ito representation

z(t) =

Z t 0

ν(dx, dt) = W (dx)dt,with W (dx) satisfying Z ∞

0 min{1, x}W (dx) < ∞,

in order for the stochastic integral in (2.29) to exist It is clear from the expression (2.29) that

z is a process with non-negative, independent and stationary increments, i.e a subordinator.Suppose z(t) is a finite activity process, then it can be written as a compound Poisson processz(t) =P N (t)

j cj, with a mean measure

ν(dx, dt) = W (dx)dt = ψP (dx ‡ c1)dt,where P (dx ‡ c1) is the probability measure and ψ is the intensity of the Poisson process

.5 1 1.5 2 25

.25 5 75 1 1.25 1.5 1.75 2 2.25

Figure 2.9: (a) rotates the IG(1,2) density for c1 (b) is the L´evy-Ito form, displaying thePoisson field N (x, t) File name is levy graphs.ox

Example 6 Figure 2.9 shows a simulation of the resulting Poisson field

N (x, t) =

Z x 0

Z t 0

N (du, ds),

in the case where c1∼ IG(δ, γ), taking ψ = 3, δ = 1 and γ = 2 We drew this by first simulating

a homogeneous Poisson process with rate ψ, and then assigning height according to draws fromthe IG(δ, γ) distribution

It is not possible to correctly draw an infinite activity process for we would have to draw aninfinite number of points in the Poisson field, although most of them would be have very littleheight

For a general L´evy process z we have the L´evy-Ito representation

t if Λ and Γ are disjoint

For a proof see, for instance, Jacod and Shiryaev (1987, Section ??)

2.7.3 Quadratic Variation

Section 2.6 showed us that it is obvious that if X is a L´evy process then its quadratic variation,[X], is also a L´evy process, in fact a subordinator An example of this is where z is a compound

In some cases it is helpful to work with an alternative, and equivalent, definition of QV which

is written in terms of a stochastic integral It is that

This is discussed in some detail in our primer on stochastic analysis For now we give an example.Example 7 Let N be a Poisson process and let us check the consistency of the formulae (2.31)and (2.28) Suppose Nt= n It is immediate from (2.28) that

[N ](t) = nwhile, on the other hand,

Suppose u(t) and v(t) are independent L´evy processes and Θ is some non-diagonal matrix.Then the linear combinations of the original L´evy processes

z(t) = Θ

(

u(t)v(t)

)

,

is a bivariate L´evy process The elements of z are marginally L´evy processes This type ofargument generalises to any dimension.

We saw in Section 2.5 that the use of subordinators can be used to generate compelling L´evyprocesses Here we use this idea to put

z(t) =

(

u(τ (t))v(τ (t))

)

,

where τ is an independent, common subordinator This means that {u(τ(t)), v(τ(t))} is now

a dependent series A concrete example of this is where {u(t), v(t)} are independent standardBrownian motions, then

z(t)|τ(t) ∼ N(0, τ(t)I),which implies the elements of z(t) are uncorrelated but are dependent In particular

Cov(z21(t), z22(t)) = Eτ (t)

2.8.2 Example: multivariate generalised hyperbolic L´evy process

Suppose we take v(t) as a d × 1 vector of correlated Brownian motions generated by

v(t) = tΣβ + Σ1/2u(t),where u is a d × 1 vector of independent, standard Brownian motions Then we take τ to be anindependent subordinator and define the deformed series

q(x) =q

δ2+ (x − µ)0Σ−1(x − µ) and α = β0Σβ

Here Σ allows us to model the correlation between the processes, while ν, δ, and γ controls thetails of the density We have a whole vector β which freely parameterises the skewness of thereturns In order to enforce identification on this model it is typical to assume that

Of course the multivariate GH density has many interesting special cases such as the multivariateStudent t, normal gamma, normal inverse Gaussian, hyperbolic and Laplace Of course thisimportant distribution can be thought of as a scale location mixture with

y = µ + Σβσ2+ σΣ1/2ε, ε ∼ N(0, I) ⊥⊥ σ2 ∼ GIG(ν, δ, γ)